Tabular Representation and Summary Statistics for One Categorical Variable

A survey asked 40 households which device they used to watch their most recent streamed show. Written out, the answers are a jumble: TV, Phone, TV, Laptop, TV, Tablet, Phone, and 33 more. Nobody can see the pattern in a list like that. Counting how many answers fall in each category turns the list into something you can read, and dividing by the total turns it into something you can compare.

Frequency tables

The frequency of a category is the number of observational units in it. A frequency table lists each category of a categorical variable next to its frequency. Here are the 40 answers, abbreviated (T = TV, P = Phone, L = Laptop, B = Tablet):

T P T L T B P T T L P T P T B T L T P T T P L T B T P T T L T P T T B P T L T P

Go through the list once, tallying each answer in its category, then check that the frequencies add up to the number of households:

The total row is the check that no answer was skipped or counted twice. Every household lands in exactly one category, so the frequencies must add to $n = 40$. Categories with no natural order can be listed in any order; ordering them from most to least frequent makes the table easier to read. If the categories have a natural order (Freshman, Sophomore, Junior, Senior, or Never, Sometimes, Often), keep that order.

Relative frequency

A count on its own doesn't tell you whether it's a lot. Ten Phone answers out of 40 is a quarter of the households; ten out of 400 would be a small minority. The relative frequency of a category is its count divided by the total:

A relative frequency table lists each category with its relative frequency:

Because the counts add to the total, the relative frequencies add to 1. When they're rounded, the sum can come out as 0.99 or 1.01; that's rounding, not an error. A sum like 0.92 or 1.10 means a category is missing or a count is wrong.

A relative frequency is a proportion, and the same information can be written several ways. The Phone category is 0.25 as a proportion or decimal, $\frac{10}{40} = \frac{1}{4}$ as a fraction, 25% as a percentage, and "1 in 4 households" in words. A ratio compares categories with each other: TV to Phone is $20 : 10$, or 2 to 1, which says TV answers were twice as common as Phone answers. Every one of these comes from the same counts, so each can be turned into the others, as long as you know which categories are involved.

There is one thing a relative frequency table can't tell you: the sample size. A table showing 0.50, 0.25, 0.15, 0.10 fits 40 households or 4,000. If you know $n$, multiply back to get counts: $\text{frequency} = \text{relative frequency} \times n$.

Using tables to back a claim

A claim about a categorical variable is supported when you can point to the specific counts or proportions that make it true. Three kinds of claim need different evidence.

Within one group. "Most households watched on a TV" needs a proportion above one half; here it is exactly 0.50, so "half" is accurate and "most" is not. "TV was the most common device" only needs TV to have the largest frequency, which it does. These are different claims: a category can be the most common with only 30% of the units, if every other category has less.

Between groups of different sizes. Compare relative frequencies, not counts. If 45 of 180 people in one town and 60 of 400 in another bike to work, the second town has more bikers but a smaller share: $\frac{45}{180} = 0.25$ against $\frac{60}{400} = 0.15$. A claim about which group bikes more often is about the proportion.

Always in context. A justification names the category, the group and the number: "25% of the 180 surveyed workers in the first town bike to work, compared with 15% of the 400 surveyed in the second." A bare "0.25 > 0.15" doesn't say what was compared.

Worked examples

Where students lose points